Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space $$X$$ in which every sequence in its continuous dual space $$X^{\prime}$$ that converges in the weak-* topology $$\sigma\left(X^{\prime}, X\right)$$ (also known as the topology of pointwise convergence) will also converge when $$X^{\prime}$$ is endowed with $$\sigma\left(X^{\prime}, X^{\prime \prime}\right),$$ which is the weak topology induced on $$X^{\prime}$$ by its bidual. Said differently, a Grothendieck space is a Banach space for which a sequence in its dual space converges weak-* if and only if it converges weakly.

Characterizations
Let $$X$$ be a Banach space. Then the following conditions are equivalent:
 * 1) $$X$$ is a Grothendieck space,
 * 2) for every separable Banach space $$Y,$$ every bounded linear operator from $$X$$ to $$Y$$ is weakly compact, that is, the image of a bounded subset of $$X$$ is a weakly compact subset of $$Y.$$
 * 3) for every weakly compactly generated Banach space $$Y,$$  every bounded linear operator from $$X$$ to $$Y$$ is weakly compact.
 * 4) every weak*-continuous function on the dual $$X^{\prime}$$ is weakly Riemann integrable.

Examples

 * Every reflexive Banach space is a Grothendieck space. Conversely, it is a consequence of the Eberlein–Šmulian theorem that a separable Grothendieck space $$X$$ must be reflexive, since the identity from $$X \to X$$ is weakly compact in this case.
 * Grothendieck spaces which are not reflexive include the space $$C(K)$$ of all continuous functions on a Stonean compact space $$K,$$ and the space $$L^{\infty}(\mu)$$ for a positive measure $$\mu$$ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).
 * Jean Bourgain proved that the space $$H^{\infty}$$ of bounded holomorphic functions on the disk is a Grothendieck space.